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A new infeasible-interior-point algorithm for linear programming over symmetric cones

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Abstract

In this paper we present an infeasible-interior-point algorithm, based on a new wide neighbourhood N(τ1, τ2, η), for linear programming over symmetric cones. We treat the classical Newton direction as the sum of two other directions. We prove that if these two directions are equipped with different and appropriate step sizes, then the new algorithm has a polynomial convergence for the commutative class of search directions. In particular, the complexity bound is O(r1.5logε−1) for the Nesterov-Todd (NT) direction, and O(r2logε−1) for the xs and sx directions, where r is the rank of the associated Euclidean Jordan algebra and ε > 0 is the required precision. If starting with a feasible point (x0, y0, s0) in N(τ1, τ2, η), the complexity bound is \(O\left( {\sqrt r \log {\varepsilon ^{ - 1}}} \right)\) for the NT direction, and O(rlogε−1) for the xs and sx directions. When the NT search direction is used, we get the best complexity bound of wide neighborhood interior-point algorithm for linear programming over symmetric cones.

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Acknowledgements

The authors are grateful to the anonymous referees and editors for their constructive and valuable suggestions improving this paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, 471023, China

    Chang-he Liu, You-lin Shang & Ping Han

  2. College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao, 266590, China

    Chang-he Liu

Authors
  1. Chang-he Liu
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  2. You-lin Shang
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  3. Ping Han
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Corresponding author

Correspondence to You-lin Shang.

Additional information

Supported by the National Natural Science Foundation of China (No. 11471102) and the Key Basic Research Foundation of the Higher Education Institutions of Henan Province (No. 16A110012).

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Liu, Ch., Shang, Yl. & Han, P. A new infeasible-interior-point algorithm for linear programming over symmetric cones. Acta Math. Appl. Sin. Engl. Ser. 33, 771–788 (2017). https://doi.org/10.1007/s10255-017-0697-7

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  • DOI: https://doi.org/10.1007/s10255-017-0697-7

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